2026/07/31 by Luca Barbieri-Viale
Physics and Astronomy · Mathematics · #quant-ph #math-ph #math.AG #math.DG #math.MP #math.QA #msc:14A15 #msc:81P16 #msc:81R05 #msc:53C80
Placed the persistent scalar trace background 1/N which defines the exact baricentro of the configuration space representing the maximally mixed state across all dimensions N
arxiv created 2026/08/03 · arxiv updated 2026/08/04
This paper establishes a rigorous, unified geometric framework for quantum state spaces by constructing smooth, regular embeddings into higher-order dual number algebras QN ≡ ℝ[ε]/(εN2-1), wherein every quantum state is faithfully represented as a non-reduced scheme-theoretic point. We show that under this unified family of truncated rings, the non-linear matrix commutators governing the Liouville-von Neumann dynamics map globally onto flat, linear, and rigid algebraic flows, establishing nilpotent dual algebras as a pristine geometric landscape for higher-dimensional quantum kinematics. As N → ∞, this family converges to a Cauchy-complete power series ring Q_∞, where non-Archimedean completion linearizes the phase space, aligning the Fubini-Study geometry with the classical Fisher-Rao manifold.